Basic trigonometric power sums with applications

Fonseca, Carlos M. da; Glasser, M. Lawrence; Kowalenko, Victor

doi:10.1007/s11139-016-9778-0

Cited by 28 publications

(20 citation statements)

References 25 publications

(33 reference statements)

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“…Now we prove our corollary. If p is a prime large enough, then, for any fixed integer , from [ 11 ] we have the identity From the orthogonality of characters and ( 17 ) we have On the other hand, from ( 13 ), ( 17 ), and Lemma 3 we also have Combining ( 18 ) and ( 19 ) we have the identity From ( 20 ) we may immediately deduce This completes the proofs of our all results.…”

Section: Proofs Of the Theoremssupporting

confidence: 51%

One kind hybrid character sums and their upper bound estimates

Zhao¹,

Xiao

2018

J Inequal Appl

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Section: Proofs Of the Theoremssupporting

confidence: 51%

One kind hybrid character sums and their upper bound estimates

Zhao¹,

Xiao

2018

J Inequal Appl

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show abstract

“…Here this dependence is avoided. This in turn allows for the determination of a summation not covered in [6,7]. It also yields representations similar in form to those in [7] for a sum analysed in [5].…”

Section: Appendix Bmentioning

confidence: 72%

“…Included here, as a last example, is a statement of the sine summation given in [6] and in [7], in a representation furnished via this method.…”

Section: As For the Previous Summation For P 2nmentioning

confidence: 99%

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Falling coupled oscillators and trigonometric sums

Holcombe

2018

Z. Angew. Math. Phys.

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A method for evaluating finite trigonometric summations is applied to a system of N coupled oscillators under acceleration. Initial motion of the nth particle is shown to be of the order  2 n 2 T for small time T and the end particle in the continuum limit is shown to initially remain stationary for the time it takes a wavefront to reach it. The average velocities of particles at the ends of the system are shown to take discrete values in a step-like manner.which is a significantly different representation to that found in [5]. While this representation offers no advantage (the representation in [5] offers a reasonably simple and excellent approximation), the method by which it was obtained appears reasonably robust in its applicability. Whereas representations for trigonometric sums, specifically the basic sine related trigonometric sums, contained within [5,6,7] were arrived at using several different methods, ranging in complexity, they may all be obtained using the very simple method discussed in the appendix.

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“…It should be emphasized that by far, the most interesting trigonometric power sums are those with inverse powers of the sine or cosine including tangent and cotangent because the evaluation of these sums generally involves the zeta function directly or through related numbers such as the Bernoulli and Euler numbers. That is, whenever the power of the sine or cosine function in the summand is even and negative, the solution is frequently number-theoretical in nature [6,7,47], whereas if it is a basic trigonometric power sum, i.e., possessing only even positive powers of sine or cosine, then it tends to be combinatorial [14]. Moreover, the inverse power case can result in the development of new and interesting results in number theory, as will be observed in this work.…”

Section: Introductionmentioning

confidence: 99%

Generalized cosecant numbers and trigonometric inverse power sums

Carlos

Glasser

Kowalenko

2018

Appl Anal Discrete Math

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The generalized cosecant numbers denoted here by c ρ,k represent the coefficients of the power series expansion or generating function of the fundamental function x ρ / sin ρ x. In actual fact, these interesting numbers are polynomials in ρ of degree k, whose coefficients are only dependent upon k. In this paper we show how they emerge in the calculation of trigonometric inverse power sums. After introducing the generalized cosecant numbers we present a novel and elegant integral approach for computing the Gardner-Fisher trigonometric inverse power sum, which is given by Sv,2(m) = π 2m 2v m−1 k=1 cos −2v kπ 2m , where m and v are positive integers. This method not only confirms the solutions obtained earlier by an empirical method, but it is also much more expedient from a computational point of view. By comparing the formulas from both methods, we derive several new and interesting number-theoretical results involving symmetric polynomials over the set of quadratic powers up to (v − 1) 2 and the generalized cosecant numbers.

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Basic trigonometric power sums with applications

Abstract: Abstract. We present the transformation of several sums of positive integer powers of the sine and cosine into non-trigonometric combinatorial forms. The results are applied to the derivation of generating functions and to the number of the closed walks on a path and in a cycle.

Cited by 28 publications

References 25 publications

One kind hybrid character sums and their upper bound estimates

One kind hybrid character sums and their upper bound estimates

Falling coupled oscillators and trigonometric sums

Generalized cosecant numbers and trigonometric inverse power sums

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